Prime Numbers in Arithmetic Progressions:
Dirichlet’s theorem and more

Sponsored by

Lecturers

Professor Timothy Trudgian, UNSW Canberra
Dr Valeriia Starichkova, UNSW Canberra
Dr Michaela Cully-Hugill, UNSW Canberra

Synopsis

We will cover several topics in Number Theory. Our main goal will be to prove the Dirichlet theorem for prime numbers in arithmetic progressions. Within the course, we will also cover other related topics. Participants will develop problem-solving skills through various exercises and motivating discussions during the workshops. Participants will get acquainted with some of the basic techniques used nowadays in analytic number theory.

Course overview

Week 1:

  • Elementary proofs about primes in arithmetic progressions.
  • Primes congruent to 1 modulo k.
  • Elementary attempts to prove the Dirichlet theorem via methods by Chebyshev and Erdős.

Week 2:

  • Obstructions of elementary methods.
  • Introduction to complex analysis and analytic techniques covering the Riemann zeta-function and its extension to the complex plane using the functional equation.
  • An analytic proof of the infinitude of prime numbers.

Week 3:

  • Introduction to the proof of the Dirichlet theorem: the Dirichlet characters and the properties of the Dirichlet L-functions.

Week 4:

  • Finalising the proof of the Dirichlet theorem.
  • More questions around primes represented by polynomials and primes in arithmetic progressions like the Green-Tao theorem.

Prerequisites

  • Arithmetic progressions; acquaintance with modular arithmetic; complex numbers; one-variable calculus.
  • A basic understanding of complex analysis would be helpful but not required.

Assessment

  • Problem-solving (60%): Students will solve exercises and present them to lecturers or workshop demonstrators during the workshops. May include some quizzes conducted in workshops.
  • Final exam at the end of week 4 (40%)

(subject to change)

Attendance requirements

Participation in all lectures and tutorials is expected.

For those completing the subject for their own knowledge/interest, evidence of at least 80% attendance at lectures and tutorials is required to receive a certificate of attendance.

Resources/pre-reading

No preliminary reading is required, but the books below will be helpful for those who would like to get acquainted with the techniques earlier.

  • “An Introduction to the Theory of Numbers” by G.H. Hardy and E.M. Wright; or
  • “Multiplicative Number Theory” by H. Davenport

Good resources for practice are:

  • “Problems in Analytic Number Theory” by M.R. Murty; or
  • “Steps into Analytic Number Theory” by P. Pollack and A. Singha Roy.

Not sure if you should sign up for this course?

Take this QUIZ to self-evaluate and get a measure of the key foundational knowledge required.

Dr Tim Trudgian
ARC Future Fellow, School of Physical, Environmental & Mathematical Sciences

Professor Timothy Trudgian, UNSW Canberra

Dr Timothy Trudgian is an associate professor at UNSW Canberra. Originally from Brisbane, he obtained a DPhil in mathematics from Oxford and returned to Canberra, after a two-year post-doc in Canada. His research interests lie in analytic number theory such as the Riemann zeta-function, distribution of primes, and primitive roots. He is a part of the Number Theory Group at UNSW Canberra and works closely with colleagues in number theory at UNSW Sydney

Dr Michaela Cully-Hugill, UNSW Canberra

Michaela Cully-Hugill is a recent PhD graduate in Number Theory from the University of New South Wales in Canberra. Her research has mainly been on explicit estimates for the distribution of primes, including error estimates for the Prime Number Theorem, and interval estimates for primes. She currently teaches at UNSW Canberra

Dr Valeriia Starichkova, UNSW Canberra

Valeriia Starichkova is finishing her PhD in number theory at UNSW Canberra in September 2023. Her research interests lie in prime numbers in short intervals, sieve theory, the analytic properties of the Riemann zeta-function and the Dirichlet L-functions, and the connections to other areas such as probabilistic or algebraic number theory.